Half-Angle Formula Calculator

The half-angle identities express sin(θ/2), cos(θ/2), and tan(θ/2) in terms of cos(θ). They are derived from the double-angle formulas by replacing θ with θ/2:

Starting from cos(2x) = 1 − 2sin²(x), replace x with θ/2: cos(θ) = 1 − 2sin²(θ/2) → sin(θ/2) = ± √((1 − cos θ) / 2)

Starting from cos(2x) = 2cos²(x) − 1: → cos(θ/2) = ± √((1 + cos θ) / 2)

For tangent, a cleaner form avoids the ± ambiguity: tan(θ/2) = sin θ / (1 + cos θ) = (1 − cos θ) / sin θ

The ± signs on sin and cos depend on the quadrant of θ/2, not θ. For example, if θ = 200°, then θ/2 = 100°, which is in the second quadrant — so sin(θ/2) is positive but cos(θ/2) is negative.

This calculator computes the actual signed values directly using JavaScript’s built-in math functions, so the signs are always correct.

When do half-angle formulas come up? Integration is the main case — specifically when you need to integrate sin²(x) or cos²(x). The power-reduction formulas (which are just half-angle formulas rearranged) turn those integrals from hard to trivial. They also appear in exact computations: sin(22.5°) = sin(45°/2) = √((1 − √2/2)/2), which simplifies to √(2 − √2)/2.

Less obvious use: exact values for angles that are half of the standard 30°/45°/60°/90° set.